I work now a lot with strings of characters and other finite sequences and found that I need many times a good notation for "cutting the end" a string. If $a$ is a finite sequence and $a'$ is its right end, then there is a sequence $x$ such that $a = x a'$ (where the product is defined by concatenation). What I would like to have is a suggestive notation for $x$: $x$ might be $a$ "minus" $a'$, or $a$ "divided by $a'$, if we view the set of all strings as a semigroup. I have therefore thought of writing $x$ as $a / a'$ but am not completely happy with it. Obvoiusly I need also a notation for "cutting at the left end"; by continuing with the previous idea the notation $a' \setminus a$ could stand for the solution of $a = a' x$, but it collides a bit with the notation for set difference.
So far I have thought, and my question is: Is there a good notation for these concepts already in use in some areas of mathematics or computer science, or has someone else already defined a suggestive notation?
P.S. And I also would like to have a notation for the "overlapping concatenation" of strings: If $a = a'x$ and $b = xb'$, what is $a'xb'$?